A New Conway Maxwell–Poisson Liu Regression Estimator—Method and Application

Poisson regression is a popular tool for modeling count data and is applied in medical sciences, engineering and others. Real data, however, are often over or underdispersed, and we cannot apply the Poisson regression. To overcome this issue, we consider a regression model based on the Conway–Maxwell Poisson (COMP) distribution. Generally, the maximum likelihood estimator is used for the estimation of unknown parameters of the COMP regressionmodel. However, in the existence of multicollinearity, the estimates become unstable due to its high variance and standard error. To solve the issue, a new COMP Liu estimator is proposed for the COMP regression model with over-, equi-, and underdispersion. To assess the performance, we conduct a Monte Carlo simulation where mean squared error is considered as an evaluation criterion. Findings of simulation study show that the performance of our new estimator is considerably better as compared to others. Finally, an application is consider to assess the superiority of the proposed COMP Liu estimator.,e simulation and application findings clearly demonstrated that the proposed estimator is superior to the maximum likelihood estimator.


Introduction
Regression models are the most popular tool for modeling the relationship between a response variable and a set of explanatory variables. In many real-life problems, the response variable is in the form of counts i.e., takes on nonnegative integer values. For count data, the most widely used regression model is the Poisson regression [1,2]. One of the major features of the Poisson distribution is that the mean and variance of the random variable are equal. However, data often exhibit over or under dispersion. In such circumstances, the Poisson distribution often does not provide good approximations. For overdispersed data, the Negative Binomial model is a popular choice [3]. Other overdispersion models include Poisson mixtures [4].
However, these models are not good for underdispersion. A flexible alternative that captures both over and underdispersion is the Conway-Maxwell Poisson (COMP) distribution, which was introduced by Conway and Maxwell in 1962 for modeling queuing systems with state-dependent service rates. e COMP distribution is a two-parameter generalization of the Poisson distribution, which also includes the Bernoulli and geometric distribution as special cases [5]. Shmueli et al. [5] established the statistical properties and parameter estimation methods of the COMP distribution.
Generally, the maximum likelihood estimation (MLE) method is a commonly used estimation method in order to estimate the unknown parameters of the COMP regression model (COMPRM). However, it is well known that the MLE is very sensitive to ill-conditioned data. Since poor estimates are produced in the existence of high but imperfect multicollinearity [11]. e major drawbacks of multicollinearity are the variance and standard error becomes high [12][13][14]. Further t and F ratios are statistically insignificant.
To reduce the effect of multicollinearity, different biased estimators are available in the literature. Among these, the most common and familiar estimation method is the Liu estimator initially introduced by Keijan [15]. For the linear regression model (LRM), we recommend the readers to see [16][17][18][19][20]. However, the literature on generalized linear model (GLMs) is limited. For detailed description, we refer, Månsson et al. [21] proposed the Liu estimator for the logit model. Månsson et al. [22] introduced some biasing parameters for the Poisson Liu estimator, Månsson [23] considered some shrinkage parameters for the negative binomial regression model. Qasim et al. [24] considered some biasing parameters for the gamma Liu regression model and Wu et al. [25] introduced the restricted almost unbiased Liu estimator for the logistic model. Amin et al. [2] studied the performance of some ridge parameter estimators in the bell regression model. Khan et al. [38] studied the influence diagnostic methods in the Poisson regression model with the Liu Estimator. Majid et al. [26] proposed some Liu parameter estimators for the bell regression model. Recently, Sami et al. [27] suggested the best ridge parameter estimator for the COMPRM. e present literature indicates that no such study related to the Liu estimator for the COMPRM is available. erefore, we propose a Liu estimator for the COMPRM to minimize the effect of collinearity among the explanatory variables. e main aim of this study is to propose a Liu estimator for the COMPRM with some new Liu parameters. For assessing the performance of these new Liu parameters, we conduct a Monte Carlo simulation study under different evaluated scenarios. e rest of the article is organized as follows: we present the statistical methodology of COMPRM in Section 2. However, a simulation layout and the results of Monte Carlo simulation are addressed in Section 3. A real-life dataset is presented in Section 4. e article ends with some concluding remarks.

Preliminaries: The COMPRM and Estimation Methods
Consider the response variable (y) comes from a COMP(λ, ϑ) with density function defined by where where λ indicates the location parameter which is the mean function of the response variable and ϑ indicates dispersion parameter. Z is the normalizing constant. ere are different indications of the dispersion parameter for different parametric values, such as if ϑ > 1, then data will be underdispersed, if ϑ < 1, then data will be overdispersed and if ϑ � 1, then data will be equally dispersed. e COMP distribution is also have relations with other distributions under different parametric conditions. For example if ϑ � 0 and λ < 1, then COMP distribution will becomes geometric distribution, if ϑ ⟶ ∞, then COMP distribution will becomes Bernoulli distribution, and if ϑ � 1, then COMP distribution will becomes the Poisson distribution. Since the COMP distribution does not have closed mathematical expressions to find its parameter so, it can be approximated by different. e asymptotic mean and variance of Y for (2) are respectively given as and Shmueli et al. [5] suggested that these approximations may not provide accurate findings when ϑ > 1 or λ 1/ϑ < 10. Regardless its flexibility and attractiveness, the COMP has restrictions in its usefulness as a basis for a Generalized Linear Model (GLM), as shown in [28,29]. In particular, neither λ nor ϑ provide a clear centering parameter. Whereas λ ≈ E(Y) when ϑ ⟶ 1, it differs substantially from the mean for small ϑ. Given that ϑ would be expected to be small for over-dispersed data, this would make a COMP model based on the original COMP formulation difficult to interpret and use for over-dispersed data [29]. So, Guikema and Goffelt [28] proposed a reparameterization using a new parameter i.e. μ � λ 1/ϑ to provide a clear centering parameter. e PMF with new formulation is defined as where By inserting μ � λ 1/ϑ in (3) and (4), the mean and variance of Y is given in terms of reparameterization as E(Y) ≈ μ + 1/2ϑ − 1/2 and Var(Y) ≈ μ/ϑ especially accurate when μ > 10 and ϑ ≤ 1. Now μ indicates the centering parameter and in new parameterization ϑ is used as a shape parameter. i.e. if ϑ < 1, the variance is greater than the mean indicating the overdispersion, however ϑ > 1 indicates underdispersion. Based on the new formulation, it is worthy to establish the GLM and by using link function and it is easier to interpret the results of coefficients [28,29]. e loglikelihood of (5) is ′ β is the linear predictor under log link, then the log-likelihood function of (7) is given by For the estimation of unknown parameters using MLE method, we use an iterative procedure. For considering the unconstrained optimization, let ϖ � log(ϑ). en, (8) becomes For finding the unknown parameters by using MLE method, we first differentiate (9) with respect to β and ϑ, respectively, as, and e estimation of β, it is required to fix ϑ. For detailed description about the information matrix, we recommend the study of Sellers and Shmueli [30]. Since COMPRM is used for modeling, the mean and variance depends on separate covariates and are, respectively, defined by [29] ln For ease, we consider a single value of ϑ. After the final iteration, the estimate of MLE becomes (13) is a feasible estimator for estimating the unknown coefficients. Fisher scoring iterative method is generally used in order to evaluate both V and z * . As the MLE has several adverse effects under multicollinearity one is that it produces larger variances. To overcome this issue, we propose the Liu estimator for the COMPRM which is called COMP Liu estimator (COMPLE) defined by where d(0 ≤ d < 1) is the Liu parameter.
where V(υ) is the covariance matrix of an estimator υ and Bias(υ) � E(υ) − υ represents the bias vector. e scalar MSE of the estimator υ is obtained by applying trace which is defined as For the comparison of two estimators υ 1 and υ 2 , the estimator υ 2 is superior to υ 1 if and only if In terms of scalar MSE, the function is true if and only if where ϑ is the dispersion parameter which is computed iteratively using (11). e MSE's of the estimators are obtained by considering α � Y t β and Λ � diag(λ 1 , λ 2 , where Y is the orthogonal matrix composed of the eigenvalues of X t VX. While α j (j � 1, . . . , p) is the jth element of Y t β. e MMSE of the β MLE is given as whereas the scalar MSE of the β MLE is e bias, covariance, and MMSE of COMPL estimator can be, respectively, computed from (14) as

Journal of Mathematics
where Λ I � diag(λ 1 + I, λ 2 + I, . . . , λ p + I) and e scalar MSE of the COMPLE is defined by where α 2 j is the jth element of Y t β MLE . Since Keijan [15] showed that the Liu estimator gives a better performance than the ordinary least squares estimator, we are extending Liu estimator for the COMPRM called the COMPLE. For this purpose, we follow Keijan [15] and differentiate (23) with respect to d, we have Putting d � 1, we have Hence there exists 0 ≤ d < 1 such that g(d) < g(1) or equivalently, MSE(β COMPLE ) < MSE(β MLE ).

Lemma 1. Let M be a positive definite (pd) matrix, α be vector of nonzero constants and c be a positive constant. en
Proof. e difference among the MMSE functions of MLE and COMPLE is obtained by However, for scalar MSE the last expression is written as After simplification, (27) can be written as us, if 0 ≤ d ≤ 1, then the proof is ended by Lemma 1. □

Selection of the Shrinkage Parameter.
e COMPLE is a better estimation method to deal with multicollinear regressors than the OLS. For the selection of optimal value of d, we follow the work of Månsson et al. [22], and differentiate (23) with respect to d and equating to zero, we have e range of d depends on α 2 j . Based on the theoretical work of [21,24,32], we define the following optimal value of d which is defined as Furthermore, Qasim et al. [33] proposed some biasing parameters which we are also considered to assess the performance i.e.
From this, following are our proposed estimators

Monte Carlo Simulation Study
is section presents a brief discussion about the generation of data with different factors that plays a crucial role in the construction of a simulation experiment. In addition, the assessment criteria are presented to examine the performance of the COMPLE with the traditional MLE.

Simulation Layout.
e response variable y i of the COMPRM is generated from a COMP (μ i , ϑ) distribution, where Following [32], the correlated regressors are generated as (33) where z ij are the independent standard normal pseudorandom numbers, and ρ 2 is the correlation between the explanatory variables. In this study, to examine the effect of different degrees of collinearity on the estimators, the following different values are considered: ρ 2 � 0.80, 0.90, 0.95, and 0.99. e slope parameters are decided such that p j�1 β 2 j � 1, which is a commonly used restriction in the field, for further details see [32]. Further, four different values of sample sizes are considered to be 50, 100, 150, and 200. e number of regressors to be included in this study are 3, 6, 9, and 12. We consider three dispersion levels i.e. for over dispersion ϑ � 0.85, for equi dispersion ϑ � 1 and for under dispersion, we consider ϑ � 1.25 to clearly monitor the performance of the proposed estimator. For the different combinations of the various values for n, p, ρ, ϑ, the data is generated 2000 times [32]. e MSE criteria is used for the evaluation of proposed and other considered estimators which is defined as where (β i − β) is the difference between the estimated and true parameter vectors of the proposed estimators at ith replication and R represents the number replications.

Results and Discussion.
e estimated MSE's of the COMPLE with the proposed Liu parameters are shown in Table 1-12. Various conditions are considered to judge the efficacy of COMPLE. e general comments on the simulation findings are discussed as follows: (1) From the provided evidences, we revealed that the overall performance of the proposed COMPLE under different shrinkage estimators is better as compared to the MLE. It can be seen that MLE is the most severely affected estimator due to its larger MSE in the presence of multicollinearity problem. (2) By fixing n, p, and ϑ, the degree of multicollinearity has a direct impact on the estimated MSE's of the COMPRM. Furthermore, by increasing the level of Journal of Mathematics

An Illustrative Example
In this section, the implementation of the proposed strategy is illustrated by a study applied to a medium-sized timber industry which manufactures laminated plastic plywood. e study consisted in evaluating the effect of explanatory variables over the number of defects found in manufactured plywood. is dataset includes n � 100 observations. We are considering the number of defects per laminated plastic plywood area (y) while four explanatory variables i.e. x 1 is the volumetric shrinkage, x 2 shows the assembly time, x 3 represents the wood density, and x 4 describes the drying temperature. We have data about the number of imperfections accompanied by the input data of the four process variables as described above. To assess the dispersion of the response variable, we use the index of dispersion (D) which is computed as D � σ 2 y /μ y [34]. e estimated value of D of the consider application found to be 135.64. As D is greater than one, which shows that, the response variable has over dispersion. Moreover, we also compute the dispersion parameter that is obtain by using (11) iteratively. Using the COMPoissonReg R package, we found that ϑ � 0.9614 which clearly demonstrates that there is over dispersion in the data set.
For assessing the multicollinearity among the considered dataset, we use condition index which is to 8634.73 > 30 which clearly indicates the presence of severe multicollinearity issue among the explanatory variables. e estimated coefficients, standard errors, and the values of MSE criterion are reported in Table 13. e estimated coefficients of the MLE and COMPLE under different shrinkage parameters are respectively obtained using (13) and (14). Whereas the scalar MSEs of the estimators are respectively computed using (21) and (23  than the COMPLE with all Liu parameters. It is of course clear that the performance of all the shrinkage parameters of the COMPLE is better as compared to the MLE. However, more specifically, the performance of the proposed COM-PLE under d 12 shows a much more robust behaviour due to its smaller values of SEs as well as the estimated MSE's. e SEs are computed by taking the square root of the diagonal elements of the variances of the estimators. e application results are also hold eorem 1 because λ j (λ j + 1) + (d + 1) > 0 for all j � 1,2,3,4.
We use other criteria, i.e., cross validation (CV) applied to the real-life data set for the assessment of the proposed method. e findings of average validation error with reference to CV method are shown in Table 13. For detailed description please see [35][36][37]. Since the CV is considered to examine the predictive performance of the estimators comprehensively. Results signify that the performance of the proposed COMPLE with all shrinkage Liu parameters is better as compared to the MLE. However, d 12 attains a minimum CV value as compared to other Liu parameters of COMPRM. So, both criteria, i.e., MSE and CV shows that the proposed estimator performs consistently better as compared to the competitors. Hence, the findings of real application are also compatible with the results of Monte Carlo simulations.

Concluding Remarks
is article proposed the Liu estimator under different shrinkage parameters for the COMPRM to deal with multicollinearity, under and over dispersion. e comparison of the MLE and COMPLE is also made via a Monte Carlo simulation and a real-life example. For the purpose of assessment, MSE is used as an evaluation criterion. Based on the findings of simulation study and real-life example, we see that the performance of our proposed estimator is comparatively better as compared to the MLE for under and over dispersion. However, more specifically, d 12 performs better in contrast to other COMPLE parameters and MLE. So, we suggest to use COMPLE with shrinkage parameter d 12 to estimate the COMPRM with multicollinearity as well as under and over dispersion.
Data Availability e data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.